Black and white.

Suppose there are $N$ balls in the urn, $b$ of which are black, $w$ white. Then the given condition means that

$\dbinom{b}{2} + \dbinom{w}{2} = \dbinom{b}{1} \dbinom{w}{1}$

$\Longrightarrow \dfrac{b(b - 1)}{2} + \dfrac{w(w - 1)}{2} = bw$

$\Longrightarrow b^{2} - b + w^{2} - w = 2bw$

$\Longrightarrow b^{2} - 2bw + w^{2} = b + w = N \Longrightarrow (b - w)^{2} = N.$

So $N$ must be a perfect square between $2014$ and $2114$ . The only possibility is $N = 45^{2} = \boxed{2025}$ . This would make

$b - w = 45, b + w = 2025 \Longrightarrow 2b = 2070 \Longrightarrow b = 1035, w = 990$

or $b = 990, w = 1035$ , if we had taken $w - b = 45$ .